Balanced systems for $\mathrm{Hom}$

TL;DR

This paper introduces finite balanced systems in homological algebra to create balanced pairs for the Hom functor, unifying various known contexts involving Gorenstein modules and relative derived functors.

Contribution

It develops the concept of finite balanced systems to systematically induce balanced pairs for Hom, extending known frameworks in relative homological algebra.

Findings

Unifies various contexts of relative Hom functors

Provides a new framework for balanced pairs in homological algebra

Applicable to Gorenstein modules and complexes

Abstract

From the notion of (co)generator in relative homological algebra, we present the concept of finite balanced system $[(\mathcal{X} , \omega ); (\nu, \mathcal{Y})]$ as a tool to induce balanced pairs $(\mathcal{X} , \mathcal{Y} )$ for the $\mathrm{Hom}$ functor with domain determined by the finiteness of homological dimensions relative to $\mathcal{X}$ and $\mathcal{Y}$. This approach to balance will cover several well known ambients where right derived functors of $\mathrm{Hom}$ are obtained relative to certain classes of objects in an abelian category, such as Gorenstein projective and injective modules and chain complexes, Gorenstein modules relative to Auslander and Bass classes, among others.